Subrings of Integers are Sets of Integer Multiples
Theorem
Let $\struct {\Z, +, \times}$ be the integral domain of integers.
The subrings of $\struct {\Z, +, \times}$ are the rings of integer multiples:
- $\struct {n \Z, +, \times}$
where $n \in \Z: n \ge 0$.
There are no other subrings of $\struct {\Z, +, \times}$ but these.
Proof
From Integer Multiples form Commutative Ring, it is clear that $\struct {n \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$ when $n \ge 1$.
We also note that when $n = 0$, we have:
- $\struct {n \Z, +, \times} = \struct {0, +, \times}$
which is the null ring.
When $n = 1$, we have:
- $\struct {n \Z, +, \times} = \struct {\Z, +, \times}$
From Null Ring and Ring Itself Subrings, these extreme cases are also subrings of $\struct {\Z, +, \times}$.
From Subgroups of Additive Group of Integers, the only additive subgroups of $\struct {\Z, +, \times}$ are $\struct {n \Z, +}$.
So there can be no subrings of $\struct {\Z, +, \times}$ which do not have $\struct {n \Z, +}$ as their additive group.
Hence the result.
$\blacksquare$
Examples
Even Integers
Let $2 \Z$ be the set of even integers.
Then $\struct {2 \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$.
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 19$. Subrings: Example $32$
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.3$: Some properties of subrings and ideals: Lemma $2.17$